Collimator and related methods

ABSTRACT

A collimator and related methods are shown and described. The collimator can be a multi-divergent-beam collimator having a plurality of inverted, ordered sections of a cone-beam collimator reassembled in a substantially reversed order relative to the ordering of the cone-beam collimator.

TECHNICAL FIELD

The present subject matter relates to nuclear medicine. In more detail,it relates to systems, methods, and uses for a collimator.

BACKGROUND

Recently manufacturers have found a large market for dedicated cardiacsingle photo emission computed tomography (SPECT) systems. Smallfield-of-view (FOV) cardiac SPECT systems have become popular due totheir compact design.

One company, Spectrum Digirad, developed dedicated cardiac SPECT systemsthat are small enough to be installed in a physician's office. Thesededicated SPECT systems have relatively small gamma cameras, which arebarely large enough to cover the heart. One feature of this system isthat the detectors remain stationary and the projections are collectedas the patient, sitting upright in a chair, is rotated.

Multi-pinhole collimation is the state-of-the-art in small animal SPECT,with the main advantage being the pinhole magnification effect, whichallows a high-sensitivity, high-resolution image to be obtained. Takingadvantage of modern large-area gamma cameras and multi-detector systems,the multi-pinhole technology is able to provide enough data for cardiacimaging without rotating the system gantry. A stationary system can takevery fast snapshots, obtaining true dynamic imaging. The stationarysystem makes patient motion correction easier, and is less expensive tobuild and maintain.

SUMMARY

One problem faced in a stationary imaging system is the lack ofsufficient view angles. In order to obtain more angular views, thepinholes are, in fact, operating in image reducing (instead ofmagnifying) mode. For a fixed pinhole aperture size, pinhole collimationprovides acceptable detection sensitivity if the object is very smalland placed very close to the pinhole; however, the detection sensitivitydecreases dramatically if the object is moved away from the pinhole. Asthe object is moved farther into the image reduction zone, where thepinhole magnification factor is less than one, the pinhole detectionsensitivity becomes worse.

In the above-referenced image reduction zone, the disclosedmulti-divergent-beam collimator may become more sensitive than thepinhole for the same specified spatial resolution. As discussed in moredetail below, one aspect of the disclosure relates to amulti-divergent-beam collimator. The collimator includes a plurality ofinverted, ordered sections of a cone-beam collimator reassembled in asubstantially reversed order relative to the ordering of the cone-beamcollimator.

In some examples, the each of the sections has substantially similardimensions. In other examples, the plurality of sections have dimensionsdifferent from others of the plurality of sections. Also, in someembodiments, a plurality of outer regions of the ordered regions havedimensions larger than a plurality of central regions.

In one example, the plurality of sections are portioned into a3-by-3-by-3 array of ordered regions. In another example, the pluralityof sections are portioned into a 2-by-3-by-2 array of ordered regions.

In another aspect, the disclosure is directed to a method ofconstructing a multi-divergent-beam collimator. The method can includepartitioning a cone-beam collimator into a plurality of ordered regions,inverting the plurality of ordered regions, and reassembling in asubstantially reversed order the inverted plurality of ordered regions.

In some examples, partitioning includes partitioning the cone-beamcollimator into regions having substantially equal dimensions orsections having different dimensions. In some case, a plurality of outerregions of the ordered regions can have dimensions larger than aplurality of central regions.

In another aspect, the disclosure is directed to a SPECT system.Included in the system is a camera having a detector and a collimator.The collimator includes a plurality of inverted, ordered sections of acone-beam collimator reassembled in a substantially reversed orderrelative to the ordering in the cone-beam collimator. The system alsoincludes a computing system that receives measurements from the cameraand processes those measurements.

In some examples, at least one of the camera, detector, and collimatorare stationary. Of course, various combinations or more than one of thesystem elements can be stationary. For example, each of the elements canbe stationary.

In some examples, the sections of the collimator have substantiallysimilar dimensions. In other examples, the regions have differentdimensions. For example, the a plurality of outer regions of the orderedregions can have dimensions larger than a plurality of central regions.

BRIEF DESCRIPTION OF THE DRAWINGS

The drawing figures depict one or more implementations in accord withthe present teachings, by way of example only, not by way of limitation.In the figures, like reference numerals refer to the same or similarelements.

FIG. 1 is a block diagram of an embodiment of a SPECT system.

FIG. 2 is a simplified block diagram of an embodiment of a camera of aSPECT system.

FIG. 3A-3E are block diagrams depicting a one-dimensional representationof an embodiment of a method of making multi-divergent-beam collimator.

FIG. 4A-4B are block diagrams depicting ordered sections of a collimatorand a reversal of that order.

FIG. 4C is a block diagram of an embodiment of a 3-by-3-by-3multi-divergence-beam collimator.

FIG. 4D is a block diagram of an embodiment of a 2-by-3-by-2multi-divergent-beam collimator.

FIG. 5A-5D are block diagrams depicting a two-dimensional representationof an embodiment of a method of making the multi-divergent-beamcollimator of FIG. 4D.

FIG. 6A is a block diagram depicting the view angles of the middle rowof the collimator of FIG. 4D.

FIG. 6B is a block diagram depicting the view angles of the top row ofthe collimator of FIG. 4D.

FIG. 6C is a depiction of a relative view angles of the collimator ofFIG. 4D.

FIG. 6D is a depiction of showing how an embodiment of the collimator ofFIG. 4D can cover the 180° view angle.

FIG. 7 depicts parameters in pinhole and divergent-beam systems used insome embodiments described herein.

FIGS. 8A-8C depict additional view-angles that are created bymulti-pinhole and multi-divergent beam systems, illustrating, in FIG. 8Cthat if a multi-divergent beam system is positioned in 3 non-overlappinglocations, it can cover over 180° of view-angles.

FIGS. 9A-9B depict simulation set-ups for both the multi-divergent beamsystem (FIG. 9A) and the multi-pinhole system (FIG. 9B).

FIGS. 10A-10D illustrate simulation results comparing angular samplingseffects and noise effects of the multi-divergent beam systems and themulti-pinhole systems.

FIGS. 11A-11B depict graphs that compare resolution and sensitivity of adivergent-beam collimator and a pinhole collimator.

DETAILED DESCRIPTION

In the following detailed description, numerous specific details are setforth by way of examples in order to provide a thorough understanding ofthe relevant teachings. However, it should be apparent to those skilledin the art that the present teachings may be practiced without suchdetails.

With reference to FIG. 1, a SPECT system 10 is shown and described. Thesystem 10 includes a gamma camera 14 and a data processing computingsystem 18. In some examples, an optional positioning element 22 isincluded. The gamma camera 14 is in communication with the computingsystem 18.

In operation, the camera 14 acquires radioisotope gamma ray photons,which are emitted from portion of a body 25. The camera 14 converts thephotons into electrical signals which represent that portion of the bodyemitting the photons.

As a result of the conversion, the electric signals are transformed intodata indicative of photon energy. In essence, the camera captures one ormore projection. The projections are fed into the computing system 18for the purpose of reconstructing an image of a spatial distribution ofa pharmaceutical substance that causes the emissions of the photonswithin the portion of the body by processing the data. The photon energyinformation is registered for the assessment of the amount of Comptonscattering that is introduced in the acquisition. The reconstruction ofan image of the portion of the body may be performed based on anyappropriate existing algorithm. For example, the ML-EM algorithm and theOS-EM algorithm can be used. Further details of these algorithms can befound in: Lange A K and Carson R: EM reconstruction algorithms foremission and transmission tomography. J. Comput. Assist. Tomogr., vol.8, pp. 306-316, 1984; Hudson H M, Hutton B F, and Larkin R: AcceleratedEM reconstruction using ordered subsets. J. Nucl. Med., vol. 33, p. 960,1991; and Hudson H M and Larki R S: Accelerated image reconstructionusing ordered subsets of projection data. IEEE Trans. Med. Imag., vol.13, pp. 601-609, 1994, the entire contents of which are hereinincorporated by reference.

With reference to FIG. 2 an example of a gamma camera 14 is shown anddescribed. In some instances, the camera 14 includes a detector 26, aphoto-multiplier 30, and collimator 34.

The detector 26 can be include at least one photon detector crystalfacing the portion of the body 25. The photon detector crystal may be inthe form of a semiconductor crystal or crystals. This crystal(s) may beselected from a first group including Cadmium-Telluride (CdTe),Cadmium-Zinc-Telluride (CeZnTe), Lead Iodine (PbI).

The photo-multiplier 30 is in communication with the detector 26. Thephoton detector crystal(s) in this case may be selected from a secondgroup including Sodium Iodine (NaI), Bismuth Germanate (BGO), YttriumOxyorthosilicate (YSO), Cerium-doped Lutetium Oxyorthosilicate (LSO) andCesium-Iodine (CsI) with solid state photo-diode or avalanchephoto-diode (APD).

The detector crystals listed above have different characteristics thatare relevant for SPECT imaging: they differ in their ability to resolvephoton energy (also termed “energy resolution”), their internal spatialresolution and their stopping power. These characteristics affect theresolution and sensitivity of the resultant images. Therefore, SPECTcameras utilizing different detector crystals will yield differentresolution, using the same reconstruction algorithm.

The detector 26, in some examples, may also be in the form of an arrayof photon detector crystals arranged in at least one row. The photondetector crystal array may be in the form of a plane or a ringsurrounding the portion of the body. For example, detector 26 may be ofthe kind used in a known per se Anger camera.

The collimator 34 is in communication with the detector 26. Thecollimator is a device capable of collimating radiation. In some cases,the collimator includes a plurality of long narrow tube in whichstrongly absorbing or reflecting walls permit only radiation travelingparallel to the tube axis to traverse the entire length. Said anotherway, the collimator 34 is a device that filters a stream of gamma raysso that only those traveling parallel to a specified direction areallowed through.

In operation, camera 14 acquires radioisotope gamma ray photons, whichare emitted from a portion the body 25. The photons pass through thecollimator 34. The gamma photons impinge the photon detector crystal 26.If the crystal is a semiconductor crystal selected from the first groupspecified above, then the crystal converts the photons into electricsignals, which are fed into the other components of the SPECT system 10for processing. Alternatively, if the crystal is selected from thesecond group specified above, i.e. is of the kind that utilizingphoto-multipliers, then the crystal converts photons into scintillationlight, which is, thereafter, transformed into electric signals byphoto-multiplier 30. These signals are processed by the computing system18 to reconstruct an image of the portion of the body 26 of interestusing know reconstruction algorithms.

As mentioned above, a stationary cardiac SPECT system is difficult todesign and manufacture. An approach that might aid in achieving a trulystationary SPECT system maybe to design multi-divergent-beam collimator.In some applications, a multi-divergent-beam collimator SPECT systemoutperforms a multipinhole system in terms of image resolution anddetection sensitivity. The performance can be characterized by thecontrast-to-noise ratio, because the detection sensitivity is inverselyrelated to the image noise. Using a multi-divergent-beam collimator canproduce a sufficient number of angular views that reconstruction of theimage is possible without, in some instances, having to rotate thecamera 14. Further, in cases where the camera 14 is positioned, thenumber of positions required to capture photon emissions from the body26 is reduced. There are numerous approaches to designing amulti-divergent-beam collimator. One approach is to design eachdivergent zone independently which usually results in a very expensivefabrication cost. A more economical solution is now discussed.

With reference to FIG. 3A-FIG. 3D, a method of constructing amulti-divergent-beam collimator is shown and described. In FIG. 3A-FIG.3D, a one-dimensional example is discussed. In FIG. 3A, a cone-beamcollimator is constructed. In FIG. 3B, the cone-beam collimator isinverted (e.g., flipped upside down) and becomes a divergent-beamcollimator. In FIG. 3C, the divergent-beam collimator is portioned intosections 38A, 38B, 38C (generally sections 38). The sections 38 areassigned an order. In FIG. 3D, the ordered sections 38 are separated. InFIG. 3E, the ordered section 38 are reassembled (e.g., glued together)in the reverse order. As shown, the resulting multi-divergent-beamcollimator produces multiple angular views of the portion of the body 26of interest.

When designing the multi-divergent-beam collimator 42, the originalconvergent-beam collimator is assumed to have a focal-length f, thensuch a converted multi-divergent-beam collimator 42 will have a commonfield-of-view that has a distance f away from the center of thecollimator. In other words, one way to design a multi-divergent-beamcollimator 42 with the center of the region of interest at a distance Bfrom the collimator, first fabricate a convergent-beam collimator thathas a focal-length B, then cut, rearrange, and glue the sections toconstruct a multi-divergent-beam collimator 42.

In more detail and with reference to FIG. 4A and FIG. 4B, sections 38are fabricated and then reassembled in reversed order. In this example,assume that the collimator has a frame of approximately 53 cm by 38 cm.The cone-beam focal length is approximately 400 mm. The hexagonal holediameter is 1.9 mm. The septa are 0.23 mm. The core thickness is 35 mm.The collimator has a resolution at 100 mm is approximately 8.1 mm. Thesensitivity of the collimator when operating in parallel mode is 334.The septa penetration at 140 keV is 0.6%. Working of these assumption,the cone-beam collimator is constructed as in FIG. 3A. Again, thecone-beam collimator is flipped upside down (e.g., inverted).

In FIG. 4A, the inverted collimator is portioned into sections 38. Asshown, the collimator is portioned into fifteen sections 38 havingsubstantially similar dimensions. Of course, the sections can havedifferent dimensions as will be described in more detail below. Asshown, the sections 38 have a square shape. However, other shapes can beused. For example, the sections 38 can be rectangular, triangular, orsome other polygonal shape. Of course, combinations of shapes an also beused. For example, a combination of squares and rectangles can beemployed. As shown in FIG. 4A, the sections 38 are assigned an order. Asshown, the sections 38 are labeled 38A-38O in a horizontal manner (e.g.,from right to left across a row). Of course, other orderings can be used(e.g., vertical assignments). After assigning the order, the sections 38are cut and then rearranged and reassembled in reverse order as shown inFIG. 4B. That is, section 38O that was previously in the bottom righthand corner is now positioned in the upper left hand corner.

With reference to FIG. 4C a 3-by-3-by-3 collimator is shown. Again, thesections 38 can have various shapes and sizes. The section 38 parametersof the collimator 42 depends on the detector size 26 and the trade-offbetween detection resolution and angular sampling. For a given detectorsize 26, the use of more view-angles correlates with more partitionedzones, which results in smaller projection images. The system 10resolution in SPECT is dominated by the collimator and the distancebetween the portion of the body of interest and the collimator. Due topoor sensitivity of SPECT, the image on the detector 26 cannot be toosmall. Assume in FIG. 4C that SPECT scanner detectors are 53 cm in thetransaxial direction. Considering the dead area around the partitionedcollimator zones, it is practical to have three zones in both thetransaxial and the axial directions, resulting in a partition similar tothat of the 3-by-3-by-3 multi-pinhole partition.

In FIG. 4D a 2-by-3-by-2 multi-divergent-beam collimator is shown anddescribed. This configuration provides additional view angles whencompared to the 3-by-3-by-3 configuration of FIG. 4C.

A method of constructing a multi-divergent-beam collimator 42 having a2-by-3-by-2 configuration is shown and described with reference to FIG.5A-5D. When compared to FIG. 3A-FIG. 3E, a substantially similar processis followed. FIG. 5A-5D show a two-dimensional example. In thisconfiguration, the top and bottom rows of the collimator 42 include twosections 38. The middle row includes three sections 38.

With reference to FIG. 6A and FIG. 6B, the additional view anglesprovided by the 2-by-3-by-2 configuration of FIG. 4D is shown anddescribed. The middle row of divergent-beam collimators 42 has threezones that provide view-angles of θ₁, 0, and −θ₁ relative to thecollimator's normal direction as in FIG. 6A, where θ₁ is calculated asθ₁=a tan⁻¹(D/(3B)). The top row has two zones and provides view anglesof θ₂ and −θ₁, where θ₂ is given as θ₂=a tan⁻¹(D/(6B)), as shown in FIG.6B. Similarly, the bottom row provides view angles of θ₁ and −θ₂. Theshown 2-by-3-by-2 configuration of the sections 38 provides view angles:θ₁, θ₂, 0, −θ₂, −θ₁, as shown in FIG. 6C. Note that at ±θ₁ the data aremeasure twice, but at different axial view angles.

In one example, assume that D=53 cm and B=40 cm. Using this assumption,θ₁=24° and θ₂=12.5°. Thus θ₁ is almost twice as large as θ₂. The viewangles are substantially uniformly sampled. Assuming that threedetectors are used with an angle of 60.5° between them, than an angularrange over 181.5° is substantially uniformly covered as shown in FIG.6D. For a short distance B, the angular coverage is larger. For example,assume that D=53 cm and B=30 cm, then θ₁=30.5°, θ₂=16.40°, and theangular coverage with three detector positions is substantially 232.2°.If D=53 cm and B=25 cm, then θ₁=35.2°, θ₂=19.5°, and the angularcoverage with three detector positions is substantially 269.7°. If twodetector positions are used, the angular coverage is 179.8°. That is, ifthe distance B can be shortened to 25 cm, it may be possible to use twodetector positions for cardiac SPECT imaging with themulti-divergent-beam collimator 42.

The most substantial problem faced in a stationary imaging system is thelack of sufficient view angles. In order to obtain more angular views,the pinholes are, in fact, operating in image reducing (instead ofmagnifying) mode. For a fixed pinhole aperture size, pinhole collimationprovides excellent detection sensitivity if the object is very small andplaced very close to the pinhole; however, the detection sensitivitydecreases dramatically if the object is moved away from the pinhole. Asthe object is moved farther into the image reduction zone, where thepinhole magnification factor is less than 1, the pinhole detectionsensitivity becomes worse. In this zone, the divergent-beam collimatorbecomes more sensitive than the pinhole for the same specified spatialresolution. For both pinhole and divergent-beam systems, the imagingsystem's field-of-view (FOV) is determined by the detector size and theobject-to-image reduction factor. If the detectors are the same and theimage reduction factors are the same, both systems have the same FOV.

In a SPECT study, the organ of interest is always assumed to be in thefield-of-view (FOV) of the gamma camera; the background and other organsmay be truncated, or not in the FOV, thus they are not measured.Dedicated systems are usually small, and data truncation happensfrequently. A potential drawback of the stationary SPECT system is thelack of a sufficient number of views. To solve this problem, the pinholeimaging system is used, operating in image reduction mode, so that manyangular views of the object can be obtained at a single detectorposition. In order for all pinholes to see the heart, the patient mustbe positioned away from the collimator, although this setup reduces theresolution and detection sensitivity.

FIG. 7 illustrates a pinhole collimator and a divergent-beam collimator.Here the assumption is made that the pinhole system has a focal-lengthf_(ph) and distance b_(ph) from the focal point to the point-of-interest(POI). Similarly, the divergent-beam system has a focal-length f_(div)and distance b_(div) from the focal point to the point-of-interest(POI). For a fair comparison, these two systems are required to have thesame image reduction factorf _(ph) /b _(ph) =f _(div) /b _(div),  (1)and that the object is the same distanceB _(ph) =B _(div) (that is, b _(ph) =b _(div) −f _(div) −L)  (2)away from the collimator.

In order to compare these two systems, a small object is placed at thePOI and required that the systems give identical spatial resolutions onthe detectors. Since the resolution of the two systems is fixed, thesuperior system provides greater geometric detection efficiency. Thesetwo systems are required to give identical detection sensitivities onthe detectors. Because the sensitivity of the two systems is fixed, thesuperior system will provide better resolution. Larger detectionsensitivity means that more gamma photons can be detected, and thisresults in lower Poisson noise in the data. Better resolution meanssmaller objects (e.g., lesions) can be resolved. The POI is furtherassumed to be on the central axis of the pinhole.

For the pinhole geometry, there are the following two relations:

$\begin{matrix}{{{Resolution}\text{:}\mspace{14mu} R_{ph}} \approx {d_{ph}\frac{f_{ph} + b_{ph}}{f_{ph}}}} & (3) \\{{{Geometric}\mspace{14mu}{Efficiency}\text{:}\mspace{14mu} g_{ph}} \approx {\frac{d_{ph}^{2}}{16b_{ph}^{2}}.}} & (4)\end{matrix}$

For the divergent-beam geometry, there are the following relations:

$\begin{matrix}{{{Resolution}\text{:}\mspace{14mu} R_{div}} \approx {d_{div}\frac{b_{div} - f_{div}}{L}\left( {1 + {\frac{1}{2}\frac{L}{f_{div}}}} \right)}} & (5) \\{{{Geometric}\mspace{14mu}{Efficiency}\text{:}\mspace{14mu} g_{div}} \approx {{K^{2}\left( \frac{d_{div}}{L} \right)}^{2}\left( \frac{f_{div} + L}{b_{div}} \right)^{2}}} & (6)\end{matrix}$where the septal thickness t is ignored for a moment, otherwise there isa

$\left( \frac{\mathbb{d}_{div}}{\mathbb{d}_{div}{+ t}} \right)^{2}$factor in g_(div).

The requirement for the two systems having the identical spatialresolution on the detectors impliesR _(ph) =R _(div),  (7)and from (3), (5) and (7), we have

$\begin{matrix}{{d_{ph}\frac{f_{ph} + b_{ph}}{f_{ph}}} = {d_{div}\frac{b_{div} - f_{div}}{L}{\left( {1 + \frac{L}{2f_{div}}} \right).}}} & (8)\end{matrix}$In order to satisfy (8) and (1) the hole-length L of the divergent-beamcollimator must satisfy

$\begin{matrix}{{L = {\frac{f_{div}}{\left( {{\beta \cdot \frac{b_{div} + f_{div}}{b_{div} - f_{div}}} - \frac{1}{2}} \right)}\mspace{14mu}\left\lbrack {{We}\mspace{14mu}{denote}\mspace{14mu}{this}\mspace{14mu} L\mspace{14mu}{as}\mspace{14mu} L_{R}} \right\rbrack}}{{{where}\mspace{14mu}\beta} = {\frac{d_{p\; h}}{d_{div}}.}}} & (9)\end{matrix}$

After the resolution is specified, equations (4) and (6) can be used tocompare their detection sensitivities as

$\begin{matrix}{\frac{g_{div}}{g_{ph}} = \frac{{K^{2}\left( \frac{d_{div}}{L} \right)}^{2}\left( \frac{f_{div} + L}{b_{div}} \right)^{2}}{\frac{d_{ph}^{2}}{16b_{ph}^{2}}}} & (10)\end{matrix}$where K is a constant that depends on the hole shape (˜0.24 for roundholes and ˜0.26 for hexagonal holes). If we assume K=0.25 then

$\begin{matrix}\begin{matrix}{\frac{g_{div}}{g_{ph}} = \frac{\left( \frac{d_{div}}{L} \right)^{2}\left( \frac{f_{div} + L}{b_{div}} \right)^{2}}{\frac{d_{ph}^{2}}{b_{ph}^{2}}}} \\{= {\left\lbrack {\frac{1}{\beta} \cdot \frac{b_{ph}\left( {f_{div} + L} \right)}{b_{div}L}} \right\rbrack^{2}.}}\end{matrix} & (11)\end{matrix}$

Pinhole and divergent-beam collimators with the same image reducingfactor can have different performances in terms of resolution. When thedivergent-beam collimator hole-length L satisfies (9), both collimatorsgive the same spatial resolution on the detectors for an object at thePOI. IfL>L _(R)  (12)the divergent-beam collimator will provide better resolution than thepinhole. Furthermore, ifL<L _(R)  (13)the pinhole collimator will provide better resolution than thedivergent-beam collimator.

The requirement that the two systems have identical detectionsensitivities on the detectors impliesg _(ph) =g _(div).  (14)From (11) and (2), (14) becomes

$\begin{matrix}\begin{matrix}{\frac{b_{ph}\left( {f_{div} + L} \right)}{\beta\; b_{div}L} = {1\mspace{14mu}{or}\mspace{14mu}\frac{\left( {b_{div} - f_{div} - L} \right)\left( {f_{div} + L} \right)}{\beta\; b_{div}L}}} \\{= 1.}\end{matrix} & (15)\end{matrix}$Solving for L from Eq. (15), results

$\begin{matrix}{L = {{\frac{\begin{matrix}{{- \left( {{2f_{div}} + {\beta\; b_{div}} - b_{div}} \right)} +} \\\sqrt{\left( {{2f_{div}} + {\beta\; b_{div}} - b_{div}} \right)^{2} + {4\;{f_{div}\left( {b_{div} - f_{div}} \right)}}}\end{matrix}}{2}\left\lbrack {{We}\mspace{14mu}{denote}\mspace{14mu}{this}\mspace{14mu} L\mspace{14mu}{as}\mspace{14mu} L_{S}} \right\rbrack}.}} & (16)\end{matrix}$

After the sensitivity is specified, equations (1), (3) and (5) can beused to compare the resolution as

$\begin{matrix}\begin{matrix}{\frac{R_{div}}{R_{ph}} = {\frac{f_{ph}}{\beta \cdot L} \cdot \frac{b_{div} - f_{div}}{f_{ph} + b_{ph}} \cdot \left( {1 + {\frac{1}{2}\frac{L}{f_{div}}}} \right)}} \\{= {\frac{1}{\beta} \cdot \frac{b_{div} - f_{div}}{b_{div} + f_{div}} \cdot {\left( {\frac{f_{div}}{L} + \frac{1}{2}} \right).}}}\end{matrix} & (17)\end{matrix}$

Pinhole and divergent-beam collimators with the same reduction factorcan have different performances in terms of detection sensitivity. When(15) is satisfied, both collimators give the same sensitivity for anobject at the POI. IfL<L _(S)  (18)the divergent-beam collimator will provide better sensitivity than thepinhole. Additionally, ifL>L _(S)  (19)the pinhole collimator will provide better sensitivity than thedivergent-beam.

Consequently, if L is chosen in the range of LR<L<LS, the divergent-beamsystem will outperform the pinhole system in both resolution andsensitivity. It can be established that 0<LR<LS is the case in someembodiments described herein, and that some embodiments can always bedesigned to have a divergent-beam imaging geometry that outperforms thepinhole system in both resolution and sensitivity.

The above conclusion is true when the pinhole system is operating in theimage reducing mode. If the pinhole system is operating in the imagemagnifying mode (as widely used in small animal imaging), thecounterpart of the divergent-beam system is the cone-beam system.

For any positive values of f_(div) and b_(div) with b_(div)>f_(div)>0,and for practical values of β>2, we have

$\begin{matrix}{{{b_{div} + f_{div}} > \frac{b_{div}}{2\beta} > \frac{b_{div} - f_{div}}{2\beta}},{i.e.},{{{\beta \cdot \frac{b_{div} + f_{div}}{b_{div} - f_{div}}} - \frac{1}{2}} > 0.}} & (20)\end{matrix}$From (9),

$\begin{matrix}{L_{R} = {\frac{f_{div}}{\left( {{\beta \cdot \frac{b_{div} + f_{div}}{b_{div} - f_{div}}} - \frac{1}{2}} \right)} > 0.}} & (21)\end{matrix}$

Here the assumption of β=d_(ph)/d_(div)>2 is true, because a typicalvalue of d_(ph) is approximately 6 mm and the typical d_(div) for anLEHR (low energy high resolution) collimator is about 1.1 mm and for anLEHS (low energy high sensitivity) collimator about 2.54 mm. This givesa typical β value of 2.36˜5.45, which is greater than 2.

Now we will show LS>LR, that is,

$\begin{matrix}{{\frac{\begin{matrix}{{- \left( {{2f_{div}} + {\beta\; b_{div}} - b_{div}} \right)} +} \\\sqrt{\left( {{2f_{div}} + {\beta\; b_{div}} - b_{div}} \right)^{2} + {4{f_{div}\left( {b_{div} - f_{div}} \right)}}}\end{matrix}}{2} > \frac{f_{div}}{{\beta \cdot \frac{b_{div} + f_{div}}{b_{div} - f_{div}}} - \frac{1}{2}}},} & (22)\end{matrix}$or equivalently

$\begin{matrix}{{\left( \frac{2f_{div}}{{\beta \cdot \frac{b_{div} + f_{div}}{b_{div} - f_{div}}} - \frac{1}{2}} \right)^{2} + {2\left( {{2f_{div}} + {\beta\; b_{div}} - b_{div}} \right)\left( \frac{2f_{div}}{{\beta \cdot \frac{b_{div} + f_{div}}{b_{div} - f_{div}}} - \frac{1}{2}} \right)} - {4{f_{div}\left( {b_{div} - f_{div}} \right)}}} < 0} & (23)\end{matrix}$which can be simplified as(−4β²+6β)f _(div) b _(div)+(−4β²+4β−1)f _(div) ²−(2 β−1)b_(div)²<0.  (24)

Since f_(div)<b_(div), if β>1.5 the left-hand-side of (23) is upperbounded by(−4β²+6β)f _(div) ²+(−4β²+4β−1)f _(div) ²(2β−1)f _(div) ²=−4f_(div)β(β−2)  (25)When β>2, the expression in (24) is negative. In other words, when β>2,LS>LR. A practical value of β is in the range of 2.36˜5.45. Therefore,in some embodiments described herein LS>LR in all cases.

The relationship 0<LR<LS assures the existence of divergent-beamcollimators that are superior to the image-reducing mode pinholecollimator in terms of both resolution and detection sensitivity.

Hereafter we address more realistic collimation situations where weconsider collimator penetration and a distributed source, which may notbe exactly at the center of the field-of-view. We assume that theradiation source is a three-dimensional cube of size 15 cm×15 cm×15 cmcontaining the heart, the collimator is made of lead, and there is anangle θ between a general emission ray and the central line of thecollimator. Based on these generalizations, equations (3)-(6) arerevised as (26)-(29):

$\begin{matrix}{{{Pinhole}\mspace{14mu}{Collimator}\mspace{14mu}{Resolution}\text{:}\mspace{14mu} R_{ph}} \approx {{\hat{d}}_{ph}\frac{f_{ph} + b_{ph}}{f_{ph}}}} & (26) \\{{{Geometric}\mspace{14mu}{Efficiency}\text{:}\mspace{14mu} g_{ph}} \approx \frac{{\hat{d}}_{ph}^{2}\cos^{3}\theta}{16b_{ph}^{2}}} & (27)\end{matrix}$where {circumflex over (d)}_(ph)=√{square root over(d_(ph)[d_(ph)+2μ⁻tan(α/2)])} is Anger's effective pinhole diameter, μis the linear attenuation coefficient of the collimator material, and αis the pinhole acceptance angle. More accurate effective pinholediameters can consider photon penetration. For large pinholes (with adiameter larger than 1 mm), Anger's effective pinhole diameter isacceptable, and will be adopted and incorporated herein for itssimplicity. Similarly, for the divergent-beam geometry, we have:

$\begin{matrix}{{{Divergent}\text{-}{Beam}\mspace{14mu}{Collimator}\mspace{14mu}{Resolution}\text{:}\mspace{14mu} R_{div}} \approx {d_{div}{\frac{b_{div} - f_{div}}{\hat{L}} \cdot \frac{1}{\cos\;\theta}}\left( {1 + {\frac{1}{2}\frac{\hat{L}}{f_{div}}}} \right)}} & (28) \\{{{Divergent}\text{-}{Beam}\mspace{14mu}{Collimator}\mspace{14mu}{Geometric}\mspace{14mu}{Efficiency}\text{:}\mspace{14mu} g_{div}} \approx {{K^{2}\left( \frac{d_{div}}{\overset{\_}{L}} \right)}^{2}\left( \frac{\mathbb{d}_{div}}{\mathbb{d}_{div}{+ t}} \right)^{2}\left( \frac{f_{div} + L}{b_{div}} \right)^{2}}} & (29)\end{matrix}$where t is the septal thickness and {circumflex over (L)}=(L−2μ⁻¹)/cos θis the effective hole-length, and K=0.26 for a hexagonal collimatorhole.

For a stationary system, one concern is the lack of sufficient viewangles. Both the multi-pinhole and multi-divergent-beam systems canprovide additional view angles at a fixed detector position. Theadditional view angles provided by these two types of systems aredifferent. We will use the 1D version to illustrate the basic principle.

The multi-pinhole geometry is shown in FIG. 8A, where the angle θ_(ph)is the additional view-angle a multi-pinhole system provides. Themaximum value of the additional view-angle can be determined by

$\begin{matrix}{\theta_{ph}^{\max} = {\tan^{- 1}\frac{\frac{D}{2} - \frac{{pf}_{ph}}{B_{ph}}}{f_{ph} + B_{ph}}}} & (30)\end{matrix}$where D is the detector size and ρ is the radius of the object ofinterest.

The multi-divergent-beam geometry is shown in FIG. 8B, where the angleθ_(div) is the additional view-angle provided by themulti-divergent-beam system. The maximum value of the additionalview-angle can be determined by

$\begin{matrix}{\theta_{div}^{\max} = {\tan^{- 1}{\frac{\frac{D}{2} - \frac{\rho\left( {b_{div} - B_{div} - L} \right)}{b_{div}}}{B_{div} + L}.}}} & (31)\end{matrix}$

As a special case, in some embodiments described herein,b_(div)=2B_(div), then (31) becomes

$\begin{matrix}{\theta_{div}^{\max} = {\tan^{- 1}{\frac{D - \rho + \frac{pL}{B_{div}}}{2\left( {B_{div} + L} \right)}.}}} & (32)\end{matrix}$

For given numerical values, the maximum value of the additionalview-angle provided by the pinhole is less than the maximum value of theadditional view-angle provided by the multi-divergent-beam system, andeach detector position using the multidivergent-beam collimator cancover approximately 60° of view angles. Therefore, three detectorpositions can acquire projections over 180° as shown in FIG. 8C. On theother hand, for the multi-pinhole system, three detector positions areless likely to cover 180°.

This analysis shows a distinct advantage of the multi-divergent-beamcollimator over the multi-pinhole collimator: The multi-divergent-beamcollimator can provide a larger view-angle range than the multi-pinholecollimator. This analysis can be readily extended to practical 2Dmulti-pinhole and multi-divergent-beam collimators. The view-angle rangein the axial direction is also larger for the multi-divergent-beamcollimator than for the pinhole collimator.

Under the assumption that the image reducing factor f_(div)/b_(div) in(1) is 0.5, f_(div)=40 cm and β=4, the value of LR from (9) isL _(R)=2f _(div)/(6β−1)=3.48 cm.

At L=LR=3.48 cm the sensitivity gain given by (11) isg_(div)/g_(ph)=2.03. This implies that when the pinhole and thedivergent-beam systems have the same spatial resolution at the center ofthe object, the divergent-beam system has a 2-fold sensitivity gain overthe pinhole system.

In some embodiments, under the assumption that the image systems satisfyassumption (2), the value of L_(S) can be directly solved from (16) asL_(S)=4.92 cm. That is, when the collimator hole-length is chosen asL_(S)=4.92 cm, both systems have the same sensitivity at the center ofthe object, while the divergent-beam system has better resolution thanthe pinhole system, with a resolution ratio R_(div)/R_(ph)=0.72. In thisnumerical example, L_(S)=4.92 cm is rather long from a practical pointof view. Thus, for a practical hole length L, the divergent-beamcollimator will have better sensitivity than the pinhole.

We now compare the divergent-beam collimator and the pinhole collimatorbased on the two assumptions expressed in Eqs. (1) and (2), and that thetwo collimators have the same reduction factor (0.5) and the samedistance (40 cm) from the center of the object to the collimator. Thesetwo requirements result in f_(div)=40 cm, b_(div)=80 cm, f_(ph)=20 cm,and b_(ph)=40 cm. The pinhole diameter is d_(ph)=6 mm, α=90°, thedivergent collimator hexagonal hole size is d_(div)=1.5 mm, and theseptal thickness is 0.23 mm. If we assume the hole size of 1.5 mm, holelength of 3.6 cm, the linear attenuation coefficient of lead at 140 keVof 21.66 per cm, then the penetration percentage is less than 6%. Using(26)-(29), a divergent-beam to pinhole resolution ratio plot and asensitivity ratio plot is illustrated in FIGS. 11A-11B. From the curves,we have the equal resolution parameter L_(R)=3.3 cm and at thishole-length the divergent to pinhole sensitivity gain is 2. From thesensitivity ratio plot, the equal sensitivity hole-length LS=4.7 cm andat this hole-length the divergent to pinhole resolution FWHM reductionfactor is 0.7.

For a multi-pinhole system setup: D=53 cm, ρ=10 cm, B_(ph)=40 cm, andf_(ph)=20 cm, the maximum value of the additional view-angle (forpinhole) is=19.7° according to (30).

For an equivalent multi-divergent-beam system setup: D=53 cm, ρ=10 cm,L=3.48 cm and B_(div)=B_(ph)=40 cm, the maximum value of the additionalview-angle (for divergent system) is=26.8° according to (32). Thus at afixed detector position, the multi-divergent-beam system can provide alarger angular range than the multi-pinhole system.

Two comparison studies compare the multi-divergent-beam andmulti-pinhole imaging systems via computer simulations. In both systems,the collimators had the same 2-3-2 partitions as shown in FIGS. 9A-9B.Each detector position provided 5 view-angles in the transaxialdirection. Each sub-detector zone was a 64×64 matrix with a pixel sizeof 1.25 mm. Three detector positions were used. Both collimators had thesame image reduction factor of 0.5. The adjacent detectors werepositioned 60° apart. The cardiac phantom had an outside radius of 6 cmand an inner radius of 5 cm. The heart-to-collimator distance was 40 cm.In projection data generation, we assumed that these two systems had thesame spatial resolution at the center of the object, which led to a2-fold sensitivity gain for the multi-divergent-beam system over themulti-pinhole system. The iterative ML-EM algorithm was used toreconstruct the images with 5 iterations. No resolution compensation wasused in image reconstruction.

In the first comparison study, computer simulated noiseless projectionswere used. The data were attenuation-less and scatter-free. The purposeof this study was to compare the angular sampling effects for bothimaging geometries. In both geometries, the detector partitions were thesame; however, their view-angles for each sub-detection-region weredifferent. It is clearly shown in FIG. 10A that 3 detector positionswith a multi-divergent-beam collimator provided satisfactory angularsampling; the short axis reconstructions appear as circular rings. Onthe other hand, the same 3 detector positions with a multi-pinholecollimator did not provide sufficient angular sampling; the circularrings became a little hexagon-like and the background has artifacts inthe shape of a star (see FIG. 10B).

In the second comparison study (see FIGS. 10C and 10D), computersimulated noisy projections were used. When the Poisson noise was addedto the projections, the sensitivity gain of 2 of the divergent-beamsystem over the pinhole system was incorporated. The purpose of thissecond study was to compare the noise effects for both imaginggeometries. A uniform spherical phantom of radius 6 cm was used so thatit was easier to calculate the noise standard deviation over the centerregion of the object. It was assumed that both systems had the samescanning time (of approximately 7 minutes with the patient cardiac Tc-99m dose). The multi-divergent-beam system had a total photon count of339439, and the multi-pinhole system had a total photon count of 155480.An inscribed cube inside the sphere was used to evaluate the mean andstandard deviation of the reconstructed image. The normalized standarddeviation (i.e. standard deviation divided by the mean) was 0.12 for thedivergent-beam system and was 0.16 for the pinhole system.

There are many approaches to designing a multi-divergent-beamcollimator. One approach is to design each divergent zone independentlywhich usually results in a very expensive fabrication cost. Someembodiments provide a novel and economical approach based on a cone-beamcollimator.

In order to illustrate the idea, we first use a one-dimensional (1D)example, where the cone-beam collimator degenerates into a fan-beamcollimator as shown in FIG. 6. First, we turn the collimator upsidedown, and the convergent-beam collimator becomes a divergent-beamcollimator. Second, we partition the collimator into multiple sections(or zones), and label them as a, b, and c. Third, we cut the sections.Fourth, we rearrange and attach them in a reversed order: c, b, and a.This procedure is illustrated and discussed previously with respect toFIGS. 3A-3E.

If the original convergent-beam collimator has a focal-length f, thensuch a converted multi-divergent-beam collimator will have a commonfield-of-view that has a distance f away from the center of thecollimator. In other words, if it is sought to design amulti-divergent-beam collimator with the center of the ROI at a distanceB from the collimator, first we need to fabricate a convergent-beamcollimator that has a focal-length B, then we cut, rearrange, and glueto construct a multi-divergent-beam collimator.

The fabrication of a practical two-dimensional (2D) collimator canfollow the same procedure as illustrated above. That is, we start with aregular cone-beam collimator of focal-length, B, then we partition andcut the collimator into sections, finally we rearrange the sections inthe reversed order and couple them together as illustrated in FIGS.5A-5B.

Those skilled in the art will recognize that the present teachings areamenable to a variety of modifications and/or enhancements. For example,although the above-described collimator is discussed for use in a SPECTsystem it can be used in other types of nuclear medical imagingapplications.

While the foregoing has described what are considered to be the bestmode and/or other examples, it is understood that various modificationsmay be made therein and that the subject matter disclosed herein may beimplemented in various forms and examples, and that the teachings may beapplied in numerous applications, only some of which have been describedherein. It is intended by the following claims to claim any and allapplications, modifications and variations that fall within the truescope of the present teachings.

1. A method of constructing a multi-divergent beam collimator, themethod comprising: partitioning a convergent cone-beam collimator into aplurality of ordered regions; separating members of the plurality ofordered regions from each other; and reassembling in a substantiallyreversed order the separated members of the plurality of orderedregions, such that the collimator is configured to act as amulti-divergent beam collimator.
 2. The method of claim 1, wherein thepartitioning comprises partitioning the cone-beam collimator intoregions having substantially equal dimensions.
 3. The method of claim 1,wherein the partitioning comprises partitioning the cone-beam collimatorinto regions having different dimensions.
 4. The method of claim 1,wherein a plurality of outer regions of the ordered regions havedimensions larger than a plurality of central regions.
 5. The method ofclaim 1, wherein the partitioning comprises portioning the cone-beamcollimator into a 3-by-3 array of ordered regions.
 6. The method ofclaim 1, wherein the partitioning comprises portioning the cone-beamcollimator into a 2-by-3-by-2 array of ordered regions.
 7. An apparatuscomprising: a plurality of ordered sections of a convergent cone-beamcollimator reassembled in a substantially reversed order relative to theordering of the sections of the cone-beam collimator, wherein thecollimator is positionable to function as a multi-divergent-beamcollimator.
 8. The apparatus of claim 7, wherein each of the sectionshas substantially similar dimensions.
 9. The apparatus of claim 7,wherein some of the plurality of sections have dimensions different fromothers of the plurality of sections.
 10. The apparatus of claim 7,wherein a plurality of outer regions of the ordered regions sectionshave dimensions larger than those of a plurality of central regions. 11.The apparatus of claim 7, wherein the plurality of sections areportioned into a 3-by-3 array of ordered regions.
 12. The apparatus ofclaim 7, wherein the plurality of sections are portioned into a2-by-3-by-2 array of ordered regions.
 13. A single photo emissioncomputed tomography (SPECT) system, comprising: a camera comprising adetector and a collimator, the collimator comprising a plurality ofordered sections of a convergent cone-beam collimator reassembled in asubstantially reversed order relative to the ordering of sections of inthe cone-beam collimator, wherein the collimator is positionable tofunction as a multi-divergent beam collimator; and a computing system incommunication with the camera, the computing system receivingmeasurements from the camera and processing the received measurements.14. The SPECT system of claim 13, wherein the camera is substantiallystationary.
 15. The SPECT system of claim 13, wherein each of thesections has substantially similar dimensions.
 16. The SPECT system ofclaim 13, wherein some of the sections have different dimensions. 17.The SPECT system of claim 13, wherein a plurality of outer regions ofthe ordered sections have dimensions larger than those of a plurality ofcentral regions.
 18. The SPECT system of claim 13, wherein the pluralityof sections are portioned into a 3-by-3 array of ordered regions. 19.The SPECT system of claim 13, wherein the plurality of sections areportioned into a 2-by-3-by-2 array of ordered regions.